Non-Gaussianity in Multi-field Stochastic Inflation with the Scaling Approximation

TL;DR

This paper addresses non-Gaussian statistics in multi-field stochastic inflation by deriving an analytic probability distribution using the scaling approximation and decomposing fluctuations into adiabatic and isocurvature components. It develops a two-field chaotic inflation model, reformulates the dynamics along and perpendicular to the classical attractor with variables $r$ and $\Theta$, and yields an analytic PDF $P_{sc}$ with quantified variances. The study demonstrates that the isocurvature sector can exhibit large non-Gaussianity and that adiabatic and isocurvature modes can be nonlinearly correlated, with the degree of non-Gaussianity depending on $n$, $\lambda_1$, $\lambda_2$, and their ratio, via $R_n$, $S_n$, and $\Gamma_n$. Connecting to observations, it relates the computed non-Gaussianity to the local $f_{NL}$ parameter, suggesting potential detectability for certain parameter ranges with Planck-scale data, while noting the need to incorporate spectral constraints and general potentials for robust inference.

Abstract

The statistics of multi-field inflation are investigated using the stochastic approach. We analytically obtain the probability distribution function of fields with the scaling approximation by extending the previous work by Amendola. The non-Gaussian nature of the probability distribution function is investigated decomposing the fields into the adiabatic and isocurvature components. We find that the non-Gaussianity of the isocurvature component can be large compared with that of the adiabatic component. The adiabatic and isocurvature components may be correlated at nonlinear order in the skewness and kurtosis even if uncorrelated at linear level.

Figures (4)

• Figure 1: Contours of the probability distribution function. Here the potential with $\lambda _1=10^{-5}$, $\lambda _2=10^{-4}$, and $n=2$ is considered. The initial time is chosen $N_*=100$ and the left, middle and right panels show the contour at the time of the e-folding $N=95$, $60$, and $20$, respectively. (See equation (28) for the definition of $N$)
• Figure 3: $A_R$ and $A_S$ as function of $N$. The dotted (dashed) curve plots $A_R$ ($A_S$) for the model with the parameters $\lambda_1=10^{-12}$ and $\lambda_2=10^{-13}$, while the solid curve plots $A_R$ with $\lambda_1=\lambda_2=10^{-12}$. Here we adopted $N_*=100$ and $n=1.5$.
• Figure 5: Same as figure 3, but with $n=3$.
• Figure 7: The ratio $A_S/A_R$ at the time $N=60$ as function of $L$ for the models $n=2,~3,~4$ (from top to bottom). Here we fixed $\lambda_1\lambda_2=10^{-24}$, and $N_*=100$ is adopted.